Theorem rnun 5136

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5133	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4923	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4929	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2250	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4729	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4729	. . 3 |- ran A = dom `' A
7		df-rn 4729	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3356	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2260	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1395   u. cun 3195   `'ccnv 4717   dom cdm 4718   ran crn 4719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by:  imaundi  5140  imaundir  5141  rnpropg  5207  fun  5496  foun  5590  fpr  5820  fprg  5821  sbthlemi6  7125  exmidfodomrlemim  7375